NEW YORK — If I were Brian Cashman or Joe Girardi, what would bug me the most about Saturday’s 13-9 loss to the Red Sox was seeing Will Middlebrooks, Jackie Bradley Jr., Ryan Lavarnway, and Xander Bogaerts occupying the 6-9 spots in the Boston order and having them go 6 for 17 with four RBIs, a home run, and six runs.

Here we are in the middle of a September pennant race and the Red Sox have four of their guys from the farm system providing that type of production. The Yankees have nothing resembling that, and are in fact a very old team, albeit a team that has lost three straight to Boston and still managed to score 25 runs. Which is why, folks, the Yankees still have a chance to make the playoffs.

Bogaerts had three RBIs and his first major league home run, and added a nice defensive play when he barehanded Robinson Cano’s slow roller in the fifth and in one motion threw out the All-Star second baseman.

A kind New Yorker threw the home run ball back so Bogaerts could have his souvenir. And what a first homer it was. The fifth-inning shot traveled an estimated 450 feet, clearing the bullpen. He jogged the bases gingerly; Bogaerts revealed that while he was hitting the homer he suffered a major cramp in the back of his left calf, so he never was able to fully appreciate the charge he put into Jim Miller’s offering. Bradley Jr., subbing for the injured Jacoby Ellsbury, was aboard.

Bogaerts’s mother, Sandra Brown, was back in Aruba celebrating her birthday and knowing her son, all of 20 years old, had helped win a very big game and hit his first major league homer. Bogaerts certainly had hoped to do something big to celebrate her birthday, and even thought to himself, “Wouldn’t it be nice to hit a home run for Mom?”

When he got to the dugout, he didn’t get the silent treatment, which some of his teammates have done to others in the past. He took his high-fives and accolades and moved on.

“To be honest, after I hit it I cramped up a little, so I wasn’t paying attention,” Bogaerts said. “As I got to first base I knew it was gone, but I was running to first paying more attention to my cramp. When I hit it . . . I wasn’t even looking at the ball. I’ll probably watch the video later. I knew I hit it good, but I didn’t know it was gone. Definitely good to get it out of the way and first major league hit out of the way. It’s always good to do those as soon as possible.

“It [the homer] was huge because I didn’t play for three or four days. I got to play on my mom’s birthday so it was special. I had to drink a lot of water afterward [to get rid of the cramp].”

Bogaerts also doubled down the left-field line in his first at-bat in the third inning and his ground out in the fourth knocked in a run.

Bogaerts, who is the youngest player (20 years, 341 days) to homer for the Red Sox since Dwight Evans (who was 20 years, 322 days old on Sept. 20, 1972), claimed, “I wasn’t sharp at all. I was bad in batting practice so I went back to my leg kick. I’ve always had a leg kick so I went back to that. I got some new bats, so I tried them out and it worked good. Give credit to the bat.”

On the barehanded play on Cano, the shortstop said, “I saw Cano hustle down the line so I didn’t know if I had a chance. So that was real good that I got him.”

Jonny Gomes, who hit a three-run homer and also made a nice play in left field to rob Lyle Overbay of extra bases, said of Bogaerts, “Big home run. I wasn’t doing that at 20, that’s for sure. That’s a big part of the ballpark. You can’t forget the kid is 20, and he’s a shortstop. There are not many shortstops with that power. He also made a couple of good defensive plays. A pretty good day at Yankee Stadium.”

Gomes added, “He might want to keep that cramp.”

Red Sox manager John Farrell acknowledged that Bogaerts hasn’t played that much since his recall, which made the afternoon even more impressive.

“Just an easy swing by Xander,” Farrell said. “What’s impressive more than anything is without everyday at-bats, to step in and double in his first at-bat, and the home run. He plays very easily defensively. Today was a glimpse of why people are so high on him as a player. Even at the early stage of his career, he’s very comfortable at this level.”

Middlebrooks, with two more hits, is up to .240 with 14 home runs. Bradley went 1 for 4 and reached base twice. Lavarnway went 1 for 5 with a double and an RBI but hit the ball well four times.

The Yankees are going to try to rebuild their team this offseason similar to the way the Red Sox did with strategically placed veteran players. What they can’t do is come up with a Bogaerts, Bradley, or a Middlebrooks, because they don’t have any of those types in their system.

Their young catchers haven’t come around as they had hoped and their young relievers such as Shawn Kelley and Preston Claiborne have hit walls. Brett Marshall pitched well in 4⅓ innings Saturday after starter David Huff allowed nine earned runs in 3⅓ innings. But Boston’s answer to Marshall, Brandon Workman, has been successful in high-leverage situations.

It was doubly good for the Red Sox — they beat up the Yankees for a third straight day and showed them a glimpse of the future, which right now, the Yankees have no answer for.